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The logo of the International Mathematical Olympiad.

The International Mathematical Olympiad (IMO) is an annual six-problem mathematical olympiad for pre-college students, and is the oldest of the International Science Olympiads.^[1] The first IMO was held in Romania in 1959. It has since been held annually, except in 1980. More than 100 countries, representing over 90% of the world's population, send teams of up to six students,^[2] plus one team leader, one deputy leader, and observers.^[3]

The content ranges from extremely difficult algebra and pre-calculus problems to problems on branches of mathematics not conventionally covered at school and often not at university level either, such as projective and complex geometry, functional equations, combinatorics, and well-grounded number theory, of which extensive knowledge of theorems is required. Calculus, though allowed in solutions, is never required, as there is a principle that anyone with a basic understanding of mathematics should understand the problems, even if the solutions require a great deal more knowledge. Supporters of this principle claim that this allows more universality and creates an incentive to find elegant, deceptively simple-looking problems which nevertheless require a certain level of ingenuity.

The selection process differs by country, but it often consists of a series of tests which admit fewer students at each progressing test. Awards are given to approximately the top-scoring 50% of the individual contestants. Teams are not officially recognized—all scores are given only to individual contestants, but team scoring is unofficially compared more than individual scores.^[4] Contestants must be under the age of 20 and must not be registered at any tertiary institution. Subject to these conditions, an individual may participate any number of times in the IMO.^[5]

The International Mathematical Olympiad is one of the most prestigious mathematical competitions in the world. In January 2011, Google sponsored €1 million to the International Mathematical Olympiad organization.^[6]

History

The first IMO was held in Romania in 1959. Since then it has been held every year except in 1980. That year, it was cancelled due to internal strife in Mongolia.^[7] It was initially founded for eastern European member countries of the Warsaw Pact, under the USSR bloc of influence, but later other countries participated as well.^[2] Because of this eastern origin, the IMOs were first hosted only in eastern European countries, and gradually spread to other nations.^[8]

Sources differ about the cities hosting some of the early IMOs. This may be partly because leaders are generally housed well away from the students, and partly because after the competition the students did not always stay based in one city for the rest of the IMO.^{[clarification needed]} The exact dates cited may also differ, because of leaders arriving before the students, and at more recent IMOs the IMO Advisory Board arriving before the leaders.^[9]

Several students, such as Zhuo Qun Song, Teodor von Burg, Lisa Sauermann, John Lian, Josh Li and Christian Reiher, have performed exceptionally well in the IMO, winning multiple gold medals. Others, such as Grigory Margulis, Jean-Christophe Yoccoz, Laurent Lafforgue, Stanislav Smirnov, Terence Tao, Sucharit Sarkar, Grigori Perelman, Ngô Bảo Châu and Maryam Mirzakhani have gone on to become notable mathematicians. Several former participants have won awards such as the Fields Medal.^[10]

Scoring and format

The examination consists of six problems. Each problem is worth seven points, so the maximum total score is 42 points. No calculators are allowed. The examination is held over two consecutive days; each day the contestants have four-and-a-half hours to solve three problems. The problems chosen are from various areas of secondary school mathematics, broadly classifiable as geometry, number theory, algebra, and combinatorics. They require no knowledge of higher mathematics such as calculus and analysis, and solutions are often short and elementary. However, they are usually disguised so as to make the solutions difficult. Prominently featured are algebraic inequalities, complex numbers, and construction-oriented geometrical problems, though in recent years the latter has not been as popular as before.^[11]

Each participating country, other than the host country, may submit suggested problems to a Problem Selection Committee provided by the host country, which reduces the submitted problems to a shortlist. The team leaders arrive at the IMO a few days in advance of the contestants and form the IMO Jury which is responsible for all the formal decisions relating to the contest, starting with selecting the six problems from the shortlist. The Jury aims to order the problems so that the order in increasing difficulty is Q1, Q4, Q2, Q5, Q3 and Q6. As the leaders know the problems in advance of the contestants, they are kept strictly separated and observed.^[12]

Each country's marks are agreed between that country's leader and deputy leader and coordinators provided by the host country (the leader of the team whose country submitted the problem in the case of the marks of the host country), subject to the decisions of the chief coordinator and ultimately a jury if any disputes cannot be resolved.^[13]

Selection process

A stage in the process of solving a problem¹ from the AIME, part of the US's selection process.

The selection process for the IMO varies greatly by country. In some countries, especially those in East Asia, the selection process involves several tests of a difficulty comparable to the IMO itself.^[14] The Chinese contestants go through a camp.^[15] In others, such as the United States, possible participants go through a series of easier standalone competitions that gradually increase in difficulty. In the United States, the tests include the American Mathematics Competitions, the American Invitational Mathematics Examination, and the United States of America Mathematical Olympiad, each of which is a competition in its own right. For high scorers in the final competition for the team selection, there also is a summer camp, like that of China.^[16]

In countries of the former Soviet Union and other eastern European countries, a team has in the past been chosen several years beforehand, and they are given special training specifically for the event. However, such methods have been discontinued in some countries.^[17] In Ukraine, for instance, selection tests consist of four olympiads comparable to the IMO by difficulty and schedule^{[clarification needed]}. While identifying the winners, only the results of the current selection olympiads are considered.^{[clarification needed]}

Awards

The participants are ranked based on their individual scores. Medals are awarded to the highest ranked participants; slightly fewer than half of them receive a medal. The cutoffs (minimum scores required to receive a gold, silver or bronze medal respectively) are then chosen so that the numbers of gold, silver and bronze medals awarded are approximately in the ratios 1:2:3. Participants who do not win a medal but who score seven points on at least one problem receive an honorable mention.^[18]

Special prizes may be awarded for solutions of outstanding elegance or involving good generalisations of a problem. This last happened in 1995 (Nikolay Nikolov, Bulgaria) and 2005 (Iurie Boreico), but was more frequent up to the early 1980s.^[19] The special prize in 2005 was awarded to Iurie Boreico, a student from Moldova, who came up with a brilliant solution to question 3, which was an inequality involving three variables.

The rule that at most half the contestants win a medal is sometimes broken if it would cause the total number of medals to deviate too much from half the number of contestants. This last happened in 2010 (when the choice was to give either 226 (43.71%) or 266 (51.45%) of the 517 contestants (excluding the 6 from North Korea — see below) a medal),^[20] 2012 (when the choice was to give either 226 (41.24%) or 277 (50.55%) of the 548 contestants a medal), and 2013, when the choice was to give either 249 (47.16%) or 278 (52.65%) of the 528 contestants a medal. In these cases, slightly more than half the contestants were awarded a medal.

Some of gold medal contestants during the IMO 2015 closing ceremony, Chiang Mai Thailand

Penalties

North Korea was disqualified for cheating at the 32nd IMO in 1991 and again at the 51st IMO in 2010.^[21] It is the only country to have been accused of cheating.

Summary

Members of the 2007 IMO Greek team.

The four perfect scorers in the 2001 IMO. From left to right: Gabriel Carroll, Reid Barton (both United States), Zhiqiang Zhang and Liang Xiao (both China).

The Bangladesh team at the 2009 IMO

Serbia's team for the 2010 IMO

Zhou Qun (Alex) Song (Canadian), the most successful IMO contestant with 5 golds and 1 bronze medal

	Venue	Year	Date	Top-ranked country[22]	Refs
1	Brașov and Bucharest	1959	July 23 – July 31	Romania	[23]
2	Sinaia	1960	July 18 – July 25	Czechoslovakia	[23]
3	Veszprém	1961	July 6 – July 16	Hungary	[23]
4	České Budějovice	1962	July 7 – July 15	Hungary	[23]
5	Warsaw and Wrocław	1963	July 5 – July 13	Soviet Union	[23]
6	Moscow	1964	June 30 – July 10	Soviet Union	[23]
7	East Berlin	1965	July 13 – July 13	Soviet Union	[23]
8	Sofia	1966	July 3 – July 13	Soviet Union	[23]
9	Cetinje	1967	July 7 – July 13	Soviet Union	[23]
10	Moscow	1968	July 5 – July 18	East Germany	[23]
11	Bucharest	1969	July 5 – July 20	Hungary	[23]
12	Keszthely	1970	July 8 – July 22	Hungary	[23]
13	Žilina	1971	July 10 – July 21	Hungary	[23]
14	Toruń	1972	July 5 – July 17	Soviet Union	[23]
15	Moscow	1973	July 5 – July 16	Soviet Union	[23]
16	Erfurt and East Berlin	1974	July 4 – July 17	Soviet Union	[23]
17	Burgas and Sofia	1975	July 3 – July 16	Hungary	[23]
18	Lienz	1976	July 2 – July 21	Soviet Union	[23]
19	Belgrade	1977	July 1 – July 13	United States	[23]
20	Bucharest	1978	July 3 – July 10	Romania	[23]
21	London	1979	June 30 – July 9	Soviet Union	[23]
	The 1980 IMO was due to be held in Mongolia. It was cancelled, and split into two unofficial events in Europe.[24]
22	Washington, D.C.	1981	July 8 – July 20	United States	[23]
23	Budapest	1982	July 5 – July 14	West Germany	[23]
24	Paris	1983	July 3 – July 12	West Germany	[23]
25	Prague	1984	June 29 – July 10	Soviet Union	[23]
26	Joutsa	1985	June 29 – July 11	Romania	[23]
27	Warsaw	1986	July 4 – July 15	Soviet Union  United States	[23]
28	Havana	1987	July 5 – July 16	Romania	[23]
29	Sydney and Canberra	1988	July 9 – July 21	Soviet Union	[23]
30	Braunschweig	1989	July 13 – July 24	China	[23]
31	Beijing	1990	July 8 – July 19	China	[23]
32	Sigtuna	1991	July 12 – July 23	Soviet Union	[23]
33	Moscow	1992	July 10 – July 21	China	[23]
34	Istanbul	1993	July 13 – July 24	China	[23]
35	Hong Kong[n 1]	1994	July 8 – July 20	United States	[23]
36	Toronto	1995	July 13 – July 25	China	[25]
37	Mumbai	1996	July 5 – July 17	Romania	[26]
38	Mar del Plata	1997	July 18 – July 31	China	[27]
39	Taipei	1998	July 10 – July 21	Iran	[28]
40	Bucharest	1999	July 10 – July 22	China  Russia	[29]
41	Daejeon	2000	July 13 – July 25	China	[30]
42	Washington, D.C.	2001	July 1 – July 14	China	[31]
43	Glasgow	2002	July 19 – July 30	China	[32]
44	Tokyo	2003	July 7 – July 19	Bulgaria	[33]
45	Athens	2004	July 6 – July 18	China	[34]
46	Mérida	2005	July 8 – July 19	China	[35]
47	Ljubljana	2006	July 6 – July 18	China	[36]
48	Hanoi	2007	July 19 – July 31	Russia	[37]
49	Madrid	2008	July 10 – July 22	China	[38]
50	Bremen	2009	July 10 – July 22	China	[39]
51	Astana	2010	July 2 – July 14	China	[40]
52	Amsterdam	2011	July 13 – July 24	China	[41]
53	Mar del Plata	2012	July 4 – July 16	South Korea	[42]
54	Santa Marta	2013	July 18 – July 28	China	[43]
55	Cape Town	2014	July 3 – July 13	China	[44]
56	Chiang Mai	2015	July 4 – July 16	United States	[45]
57	Hong Kong	2016	July 6 – July 16	United States	[46]
58	Rio de Janeiro	2017	July 12 – July 23	South Korea	[47]
59	Cluj-Napoca	2018	July 3 – July 14	United States	[48]
60	Bath	2019	July 11 – July 22		[49]
61	Saint Petersburg	2020			[50][51]
62		2021			[52]
63	Oslo	2022			[53][54]

Notable achievements

The following nations have achieved the highest team score in the respective competition:

• 
  China, 19 times (from the first participation in 1985 until 2014): in every year from 1989 to 2014 except 1991, 1994, 1996, 1998, 2003, 2007, 2012;


• 
  Soviet Union, 14 times: in 1963, 1964, 1965, 1966, 1967, 1972, 1973, 1974, 1976, 1979, 1984, 1986, 1988, 1991;


• 
  United States, 7 times: in 1977, 1981, 1986, 1994, 2015, 2016, 2018;


• 
  Hungary, 6 times: in 1961, 1962, 1969, 1970, 1971, 1975;


• 
  Romania, 5 times: in 1959, 1978, 1985, 1987, 1996;


• 
  West Germany, twice: in 1982 and 1983;


• 
  Russia, twice: in 1999 and 2007;


• 
  South Korea, twice: in 2012 and 2017;


• 
  Bulgaria, once: in 2003;^[55]
  


• 
  Iran, once: in 1998;


• 
  East Germany, once: in 1968.

The following nations have achieved an all-members-gold IMO with a full team:

• 
  China, 11 times: in 1992, 1993, 1997, 2000, 2001, 2002, 2004, 2006, 2009, 2010 and 2011.^[56]
  


• 
  United States, 3 times: in 1994, 2011, and 2016.^[57]
  


• 
  Russia, 2 times: in 2002 and 2008.^[58]
  


• 
  South Korea, twice: in 2012 and 2017.^[59]
  


• 
  Bulgaria, once: in 2003.^[60]
  

The only countries to have their entire team score perfectly in the IMO were the United States in 1994 (they were coached by Paul Zeitz); and Luxembourg, whose 1-member team had a perfect score in 1981. The US's success earned a mention in TIME Magazine.^[61] Hungary won IMO 1975 in an unorthodox way when none of the eight team members received a gold medal (five silver, three bronze). Second place team East Germany also did not have a single gold medal winner (four silver, four bronze).

Several individuals have consistently scored highly and/or earned medals on the IMO: As of July 2015 Zhuo Qun Song (Canada) is the most successful participant^[62] with five gold medals (including one perfect score in 2015) and one bronze medal.^[63]Reid Barton (United States) was the first participant to win a gold medal four times (1998-2001).^[64] Barton is also one of only eight four-time Putnam Fellows (2001–04). Christian Reiher (Germany), Lisa Sauermann (Germany), Teodor von Burg (Serbia), and Nipun Pitimanaaree (Thailand) are the only other participants to have won four gold medals (2000–03, 2008–11, 2009–12, 2010–13, and 2011–14 respectively); Reiher also received a bronze medal (1999), Sauermann a silver medal (2007), von Burg a silver medal (2008) and a bronze medal (2007), and Pitimanaaree a silver medal (2009).^[65]Wolfgang Burmeister (East Germany), Martin Härterich (West Germany), Iurie Boreico (Moldova), and Lim Jeck (Singapore) are the only other participants besides Reiher, Sauermann, von Burg, and Pitimanaaree to win five medals with at least three of them gold.^[2]Ciprian Manolescu (Romania) managed to write a perfect paper (42 points) for gold medal more times than anybody else in the history of the competition, doing it all three times he participated in the IMO (1995, 1996, 1997).^[66] Manolescu is also a three-time Putnam Fellow (1997, 1998, 2000).^[67]Eugenia Malinnikova (Soviet Union) is the highest-scoring female contestant in IMO history. She has 3 gold medals in IMO 1989 (41 points), IMO 1990 (42) and IMO 1991 (42), missing only 1 point in 1989 to precede Manolescu's achievement.^[68]

Terence Tao (Australia) participated in IMO 1986, 1987 and 1988, winning bronze, silver and gold medals respectively. He won a gold medal when he just turned thirteen in IMO 1988, becoming the youngest person at that time^[69] to receive a gold medal (a feat matched in 2011 by Zhuo Qun Song of Canada). Tao also holds the distinction of being the youngest medalist with his 1986 bronze medal, alongside 2009 bronze medalist Raúl Chávez Sarmiento (Peru), at the age of 10 and 11 respectively.^[70] Representing the United States, Noam Elkies won a gold medal with a perfect paper at the age of 14 in 1981. Note that both Elkies and Tao could have participated in the IMO multiple times following their success, but entered university and therefore became ineligible.

The current ten countries with the best all-time results are as follows:^[71]

Rank	Country	Appearance	Gold	Silver	Bronze	Honorable Mentions
1	China	32	147	33	6	0
2	United States	43	119	111	29	1
3	Russia	26	92	52	12	0
4	Hungary	57	81	160	95	10
5	Soviet Union[n 2]	29	77	67	45	0
6	Romania	58	75	141	100	4
7	South Korea	30	70	67	27	7
8	Vietnam	41	59	109	70	1
9	Bulgaria	58	53	111	107	10
10	Germany	40	49	98	75	11

Media coverage

• A documentary, "Hard Problems: The Road To The World's Toughest Math Contest" was made about the United States 2006 IMO team.^[72]
  


• A BBC documentary titled Beautiful Young Minds aired July 2007 about the IMO.


• A BBC fictional film titled X+Y released in September 2014 tells the story of an autistic boy who took part in the Olympiad.

See also

• List of International Mathematical Olympiads


• 
  International Mathematics Competition for University Students (IMC)


• International Science Olympiad


• List of mathematics competitions


• Pan-African Mathematics Olympiads


• Junior Science Talent Search Examination

Notes

Citations

References

• 
  Xu, Jiagu (2012). Lecture Notes on Mathematical Olympiad Courses, For Senior Section. World Scientific Publishing. ISBN 978-981-4368-94-0.
  

• 
  Xiong, Bin; Lee, Peng Yee (2013). Mathematical Olympiad in China (2009-2010). World Scientific Publishing. ISBN 978-981-4390-21-7.
  

• 
  Xu, Jiagu (2009). Lecture Notes on Mathematical Olympiad Courses, For Junior Section. World Scientific Publishing. ISBN 978-981-4293-53-2.
  

• 
  Olson, Steve (2004). Count Down. Houghton Mifflin. ISBN 0-618-25141-3.
  


• 
  Verhoeff, Tom (August 2002). "The 43rd International Mathematical Olympiad: A Reflective Report on IMO 2002" (PDF). Computing Science Report, Faculty of Mathematics and Computing Science, Eindhoven University of Technology, Vol. 2, No. 11.
  


• 
  Djukić, Dušan (2006). The IMO Compendium: A Collection of Problems Suggested for the International Olympiads, 1959-2004. Springer. ISBN 978-0-387-24299-6.
  


• 
  Lord, Mary (July 23, 2001). "Michael Jordans of math - U.S. Student whizzes stun the cipher world". U.S. News & World Report. 131 (3): 26.
  


• 
  Saul, Mark (2003). "Mathematics in a Small Place: Notes on the Mathematics of Romania and Bulgaria" (PDF). Notices of the American Mathematical Society. 50: 561–565.
  


• 
  Vakil, Ravi (1997). A Mathematical Mosaic: Patterns & Problem Solving. Brendan Kelly Publishing. p. 288. ISBN 978-1-895997-28-6.
  


• 
  Liu, Andy (1998). Chinese Mathematics Competitions and Olympiads. AMT Publishing. ISBN 1-876420-00-6.
  

External links

• Official IMO web site


• Old central IMO web site